Problem 34
Question
\(10 y^{2}=180\)
Step-by-Step Solution
Verified Answer
\[ y = 3 \sqrt{2} \quad \text{or} \quad y = -3 \sqrt{2} \]
1Step 1: Simplify the Equation
Start by dividing both sides of the equation by 10 to simplify it. \ \[ \frac{10y^{2}}{10} = \frac{180}{10} \] This simplifies to \[ y^{2} = 18 \]
2Step 2: Solve for y
Take the square root of both sides of the equation to solve for y. Remember that the square root has both positive and negative solutions. \ \[ y = \sqrt{18} \quad \text{or} \quad y = -\sqrt{18} \]
3Step 3: Simplify the Square Root
Simplify the square root of 18. Notice that 18 can be written as \sqrt{9 \times 2}. \[ \sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \sqrt{2} = 3 \sqrt{2} \] Thus, \[ y = 3 \sqrt{2} \quad \text{or} \quad y = -3 \sqrt{2} \]
Key Concepts
Simplifying EquationsTaking Square RootsPositive and Negative Solutions
Simplifying Equations
To solve quadratic equations like \(10y^2 = 180\), the first step is to simplify the equation. Simplifying equations often involves making them easier to handle or working with smaller numbers. Begin by dividing both sides of the equation by 10. This is because the coefficient of \(y^2\) is 10. By doing this, we isolate \(y^2\):
\[ \frac{10y^2}{10} = \frac{180}{10} \]
Simplifying both sides gives us:
\[ y^2 = 18 \]
The equation is now simplified and easier to solve. It's important to always check if you can reduce the numbers, as it makes the next steps more manageable.
\[ \frac{10y^2}{10} = \frac{180}{10} \]
Simplifying both sides gives us:
\[ y^2 = 18 \]
The equation is now simplified and easier to solve. It's important to always check if you can reduce the numbers, as it makes the next steps more manageable.
Taking Square Roots
The next step in solving \( y^2 = 18 \) is to take the square root of both sides. This helps us find the actual value(s) of \( y \). When we take the square root of a number, we are essentially finding a value that, when squared, gives us the original number. Therefore, we proceed as follows:
\[ y^2 = 18 \]
Taking the square root of both sides, we get:
\[ y = \text{±} \sqrt{18} \]
Remember, the square root symbol (√) represents both positive and negative solutions. This is because both positive and negative numbers when squared will produce the same result. For this problem:
\[ y = \sqrt{18} \quad \text{or} \quad y = -\sqrt{18} \]
In general, always remember that taking a square root involves considering both the positive and negative possibilities.
\[ y^2 = 18 \]
Taking the square root of both sides, we get:
\[ y = \text{±} \sqrt{18} \]
Remember, the square root symbol (√) represents both positive and negative solutions. This is because both positive and negative numbers when squared will produce the same result. For this problem:
\[ y = \sqrt{18} \quad \text{or} \quad y = -\sqrt{18} \]
In general, always remember that taking a square root involves considering both the positive and negative possibilities.
Positive and Negative Solutions
When solving equations involving square roots, it is essential to consider both positive and negative solutions. For the specific case of \( y = \sqrt{18} \), we derived two solutions:
\[ y = \sqrt{18} \quad \text{and} \quad y = -\sqrt{18} \]
But often, we need to simplify the square root to get a more precise answer. The square root of 18 can be broken down into simpler components:
\[ \sqrt{18} = \sqrt{9 \times 2} \]
We know that 9 is a perfect square, so we can further simplify:
\[ \sqrt{9} \cdot \sqrt{2} = 3\sqrt{2} \]
Therefore, we have:
\[ y = 3\sqrt{2} \quad \text{or} \quad y = -3\sqrt{2} \]
Always remember to include both positive and negative solutions in your final answer. This ensures a complete and accurate solution. In summary, for the equation \(10y^2 = 180\), the solutions are \(y = 3\sqrt{2}\) and \(y = -3\sqrt{2}\).
\[ y = \sqrt{18} \quad \text{and} \quad y = -\sqrt{18} \]
But often, we need to simplify the square root to get a more precise answer. The square root of 18 can be broken down into simpler components:
\[ \sqrt{18} = \sqrt{9 \times 2} \]
We know that 9 is a perfect square, so we can further simplify:
\[ \sqrt{9} \cdot \sqrt{2} = 3\sqrt{2} \]
Therefore, we have:
\[ y = 3\sqrt{2} \quad \text{or} \quad y = -3\sqrt{2} \]
Always remember to include both positive and negative solutions in your final answer. This ensures a complete and accurate solution. In summary, for the equation \(10y^2 = 180\), the solutions are \(y = 3\sqrt{2}\) and \(y = -3\sqrt{2}\).